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Metamath Proof Explorer

Theorem bnj529

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypothesis bnj529.1 
|- D = ( _om \ { (/) } )
Assertion bnj529 
|- ( M e. D -> (/) e. M )

Proof

Step Hyp Ref Expression
1 bnj529.1 
 |-  D = ( _om \ { (/) } )
2 eldifsn 
 |-  ( M e. ( _om \ { (/) } ) <-> ( M e. _om /\ M =/= (/) ) )
3 2 biimpi 
 |-  ( M e. ( _om \ { (/) } ) -> ( M e. _om /\ M =/= (/) ) )
4 3 1 eleq2s 
 |-  ( M e. D -> ( M e. _om /\ M =/= (/) ) )
5 nnord 
 |-  ( M e. _om -> Ord M )
6 5 anim1i 
 |-  ( ( M e. _om /\ M =/= (/) ) -> ( Ord M /\ M =/= (/) ) )
7 ord0eln0 
 |-  ( Ord M -> ( (/) e. M <-> M =/= (/) ) )
8 7 biimpar 
 |-  ( ( Ord M /\ M =/= (/) ) -> (/) e. M )
9 4 6 8 3syl 
 |-  ( M e. D -> (/) e. M )